arXiv:1307.8062v1 [astro-ph.CO] 30 Jul 2013 2Spebr2018 September 12 o.Nt .Ato.Soc. Astron. R. Not. Mon. oncaKirk Donnacha Modified and Energy Dark Gravity for Benefits Galaxy Same-sky Photometric Surveys: and Spectroscopic Optimising uvy fglxe n aaycutr Esnti 2005; (Eisenstein clusters galaxy volume su- Large and Universe. Ia galaxies the type of chart of surveys the which expansion 1999) to accelerating Perlmutter the 2013) 1998; (Riess Sievers (SNe) pernovae 2011; Cosmic the Carlstrom al. et from most 2013; Collaboration the Universe (Planck of (CMB) our Background some Microwave of measure Different properties to reality. able fundamental a now are is probes cosmology” cosmological “precision of era The INTRODUCTION 1
iieB Abdalla B. Filipe c 3 2 1 4 4 ntttd iece elEpi(C,IE/SC,E-08193 IEEC/CSIC), and (ICE, Physics l’Espai of Ci`encies de School de Astrophysics, Lond Institut for College Centre University Bank Astronomy, Jodrell & Physics of Department al nttt o omlgclPyis h nvriyo University The Physics, 605 Cosmological IL for Batavia, Institute Astrophysics, Kavli Particle for Center Fermilab 09RAS 2009 1 frLahav Ofer , 000 1 ohaA Frieman A. Joshua , –1(09 rne 2Spebr21 M L (MN 2018 September 12 Printed (2009) 1–21 , rdc outrslswihgv odaeaepcueo h sc the as of fiducial words: picture Key our average surveys. that good spectroscopic that a and give show m photometric which We of combining results analysis. figures assumpt robust forecasts sensitivity forecast of produce exact a range a The in by gravity. affected methodically modified radically e be dark for can for 2 4 benefits of of cross- matr factor these factor a covariance nearly of a full find inclusion We and surveys. the vector non-overlapping to from data t joint factor, when s observable improvement a power cross-correlation same-sky producing the angular sky, model the projected n to on Our merit easy it alone. of makes whe either figure gravity, ism The joint than modified non-overlapping alone. larger for survey a magnitude strongest either producing by are orthogonal, surveys produced are figu that the than energy between mod larger gies and dark times energy a 4 dark sees than both combination measure Com non-overlapping to surveys. a independent, power Sumire) DES their results 4MOST, KiDS, improves our (e.g. DESI, greatly redshift that (e.g. photometric redshift so future w spectroscopic designs and survey current a survey of from generic photomet range distortions choose with late-t space We survey redshift of a redshifts. and probes from clustering correlated galaxy lensing highly (ii) gravitational two weak of (i) an combination growth: with the possible those examine than We stringent measur more produce much can parameters probes logical cosmological multiple of combination The ABSTRACT rvt omlgclprmtr ag-cl tutr fUniv of structure large-scale – parameters cosmological – gravity 1 aa Bridle Sarah , omlg:osrain rvttoa esn akeeg mo – energy dark – lensing gravitational – observations cosmology: hcg,Ciao L60637 IL Chicago, Chicago, f 10 srnm,TeUiest fMnhse,Mnhse,M39 M13 Manchester, Manchester, of University The Astronomy, eltra(acln) Spain (Barcelona), Bellaterra 4 n oe tet odn CE6T UK 6BT, WC1E London, Street, Gower on, et.W ealanme fteesresi als1and 1 to 2012; tables Amiaux up in 2012; surveys ob- Collaboration scaled these DES more of & are and (Soares-Santos number volumes 2 probes a greater detail sky, cosmic We the jects. these on area of more many cover as data invisible species, ac- matter light, observation. dominant of direct the to bending matter, the dark through to cess us, gives 2008; and 2012) Jain structure & Heymans cosmic (Hoekstra (WGL) of Lensing growth Gravitational Weak the chronicle 2003) Colless 2 tpai Jouvel Stephanie , h etdcd ilbiga vngetrwv of wave greater even an bring will decade next The A T E tl l v2.2) file style X 3 , o,wihw explore we which ion, eg n oethan more and nergy orltos relative correlations, erse eterconstraints their re S,Eci)and Euclid) HSC, , niiulprobe. individual y iigtesurveys the bining esresoverlap surveys he eapial oa to applicable re x ecluaea calculate We ix. rtadsame-sky and erit ec eunfrom return ience eo ei more merit of re t spectroscopic ith al resof orders 2 early cosmo- of ements etu formal- pectrum and redshifts ric fidgaiy An gravity. ified oeflsyner- powerful m structure ime sumptions L UK PL, dified 2 Donnacha Kirk, Ofer Lahav, Sarah Bridle, Stephanie Jouvel, Filipe B. Abdalla, Joshua A. Frieman
Abdalla 2012; de Jong 2012; Pilachowski 2012; Sugai 2012a). ing RSDs) from the spectroscopic redshift (spec-z) survey. Each probe of cosmology requires an enormous effort to un- This pared down approach allows us to explore the impact derstand both the underlying physics and subtle systematic of nuisance parameter modeling & choice, survey strategy and observational effects as well as the creation of innovative and survey overlap in a clean way without having to deal new statistical techniques to deal with the sheer quantity of with too many competing effects. For the same reason we data being produced. In engineering terms these projects are choose to model both probes and their cross-correlations in often pushing boundaries in terms of space science, optics, the same projected angular power spectrum formalism. detector design, computation and data storage. Cosmologi- The combination of a photo-z WGL survey and a spec-z cal probes are generally complementary, in that each probes galaxy clustering survey has been studied by a number of a different combination of the cosmological parameters we papers including Cai & Bernstein (2012); Gazta˜naga et al. are interested in, while being sensitive to different sets of (2012); Duncan et al. (2013); de Putter et al. (2013). In gen- nuisance parameters and systematics. eral these papers have modelled different observables us- While each different cosmological probe will gather data ing different formalisms. Our approach in this paper is to of unprecedented precision over the next decade and be- model both WGL and galaxy clustering, including RSDs as yond, it is already clear that the strongest constraints on projected angular power spectra, C(l) (Hu 1999; Bernstein cosmology come from the proper combination of different 2009). While there may be some loss in accuracy for the probes (Kilbinger 2013; Jee et al. 2013). These combinations spec-z case due to projection along redshift we are inter- break degeneracies between cosmological (and nuisance) pa- ested in presenting a unified framework in which each ob- rameters and allow a level of precision much beyond any servable is treated on the same footing and cross-correlations individual probe. Indeed this is the source of our current can be handled naturally. This fits with the philosophy of “concordance cosmology”, ΛCDM (Komatsu 2011). Some jointly modelling all cosmological/systematic effects in the cosmological probes are relatively independent, perhaps the same ‘combined probes’ data vector and a single joint co- CMB and SNe are a good example. These probes can be variance matrix. In the same spirit we try to make explicit combined in a very simple way without worrying about the all assumptions about observable/survey modelling and the cross-talk between observables or double counting of infor- treatment of nuisance parameters. For the most fundamen- mation. This, however, is the exception. Most probes are tal assumptions we examine the impact of varying each in- highly correlated as they probe the same underlying phys- dependently as a sensitivity analysis. A full “optimisation” ical processes, whether that is the expansion history of the would vary these assumptions simultaneously and search for Universe or the perturbations of the large-scale gravitational the best combination but we think many are currently so potential as it evolves with time. ill-understood that it is more important to disentangle the Given this situation, increasing attention is being paid separate effects. Each assumption will require specialist at- to the correct way to combine multiple cosmological probes. tention to settle on a “correct” approach, we hope merely While the relatively independent probes we mentioned can to demonstrate the power of these assumptions to change be treated separately and combined on the level of multiplied survey results and the need for detailed further attention. posterior probabilities, this is not possible with the late- This paper forms a companion piece to Jouvel (2013). time Large-Scale Structure (LSS) probes which are highly We model similar surveys but, as a division of labour, we correlated both in terms of cosmological information and restrict consideration of target selection, survey design and in systematic effects. In these cases it is essential to con- observing strategy to Jouvel (2013). This paper considers struct a joint data vector which can model all cosmolog- assumptions on theoretical formalism, systematics includ- ical and systematic effects simultaneously, including their ing galaxy bias & photo-z error, survey overlap and more. cross-correlations, and avoid double counting. In addition Assumptions varied in Jouvel (2013) are fixed in this paper one should perform a simultaneous joint likelihood analysis and vice versa. using a covariance matrix which includes all cross correla- Section 2 talks about the landscape of photo-z and spec- tion terms (and off-diagonal elements) between the different z surveys. Then in section 3 we present our C(l)s formalism probes. If these complications are ignored the final result can for both cosmic shear and galaxy clustering before detail- be strongly biased (Eifler et al. 2013; Taylor et al. 2013). ing the rest of our assumptions about nuisance parameters For clarity this paper concentrates on the combination and fiducial survey strategies in section 4. Our forecast con- of two types of survey which will become available over straints on DE and MG are given in section 5, where each the next 5-10 years. We choose a large area optical cosmic subsection details the impact of a move away from our fidu- shear survey with photometric-quality redshifts, modelled cial assumptions. We draw together the implications of these on the Dark Energy Survey (DES) (5000 deg2 with 200 results in section 6 before concluding in section 7. ∼ million galaxies) and a medium scale spectroscopic LSS sur- vey (5000 deg2 targeting 10 million galaxies) similar to the ∼ DESI (combined Big-BOSS, DESpec), 4MOST and Sumire 2 PHOTOMETRY & SPECTROSCOPY concepts (Abdalla 2012; de Jong 2012; Pilachowski 2012; Sugai 2012a). See tables 1 and 2 for more details on cur- When we make a survey of galaxies in the Universe, rent and future surveys. Although there are many possible whether to study Redshift Space Distortions (RSDs), Weak analyses one can make with the wealth of data provided Gravitational Lensing (WGL), Baryon Acoustic Oscillations by these two types of survey (DES alone combines informa- (BAOs) or galaxy clustering itself, we need to characterise tion from WGL, LSS, galaxy clusters and SNe) we choose the position of each galaxy using three coordinates. Two of to limit ourselves to WGL from the photometric redshift these (commonly RA & DEC) locate the galaxy in two di- (photo-z) survey and LSS (galaxy power spectrum includ- mensions on the plane of the sky. It is relatively straightfor-
c 2009 RAS, MNRAS 000, 1–21 Optimising Spectroscopic & Photometric Galaxy Surveys: Same-sky Benefits for Dark Energy & Modified Gravity 3 ward to achieve a precise measurement of sky position, with cross-calibration techniques for photo-z testing (Zhang et al. accuracies of sub-arcsecond achievable even for ground based 2010), and the cross-correlation between surveys to pro- observations. In contrast, fixing the third coordinate, the vide clustering measurements that are robust to systemat- galaxy’s distance from the observer along the line of sight, ics (Yoo & Seljak 2012). Detailed discussions on these issues is considerably more challending. We measure a galaxy’s red- are given in both the BigBOSS and DESpec white papers shift, the lengthening of the wavelength of light from that (Abdalla 2012; Pilachowski 2012), and in the report of the galaxy as it recedes from us under the Hubble flow, and use Joint Working Group BigBOSS-DES1. it to determine distance. More distant objects have greater There are a number of suitable photo-z and spec-z sur- redshifts. In general a measurement of distance requires as- veys already available with many more in progress or due to sumptions to be made about cosmological parameters while start over the coming years on different parts of the sky. As a redshift measurement does not. results from more surveys become available, the optimal use The most accurate method for determining a galaxy’s of overlapping photo-z/spec-z sky area will become a crucial redshift is spectroscopy. Light from the galaxy is split into question if we are to obtain the best constraints on cosmol- its frequency components and the movement of spectral fea- ogy from the available data. We summarise some current tures to the red is used to measure an accurate redshift. and future spectroscopic surveys in table 1 and do the same Spectroscopic redshifts (spec-z) can reach an accuracy of for photometric surveys in table 2. − better than 10 3, however this process is costly and time It has been illustrated in a number of pa- consuming. Each galaxy must be examined individually and pers (Bernstein & Cai 2011; Cai & Bernstein 2012; observed for sufficient time that enough light is collected Gazta˜naga et al. 2012; Abdalla 2012; Pilachowski 2012; and a clear spectrum established. Modern multi-object spec- Duncan et al. 2013) that a combination of Redshift Space trographs expedite this process by using multiple optical fi- Distortion (RSD) from spectroscopic surveys and weak bres to collect light for up to 4,000 galaxies simulataneously. lensing from imaging surveys is a very powerful tool However even these cutting edge, high-throughput machines to constrain Dark Energy and deviations from General are limited to observing 60,000 galaxy spectra per observ- Relativity. Weak lensing and BAO/RSD are unique probes ∼ ing night (Abdalla 2012). of large-scale structure, exploring different scales in k-space, There is a faster but less accurate redshift estimation where a combined analysis may be able to remove some technique in common use for large optical surveys. Known underlying degeneracies. as photometric redshift (photo-z) estimation, it dispenses Furthermore, having both the spectroscopy and imag- with the spectrograph entirely and relies on the fact that a ing on the same part of sky could provide access to new standard optical survey will observe in multiple frequency cosmological tests. When observed on the same part of the bands (u, g, r, i, z, y etc.), recording images for each expo- sky, the galaxies observed with a spectroscopic survey map sure under each filter on a many mega-pixel CCD camera. the underlying mass fluctuations that lead in turn to the Combining intensity information for a single object from weak lensing of the distant imaged galaxies. This constrains multiple filters produces what is in effect a very low res- directly galaxy biasing (Gazta˜naga et al. 2012), reduces cos- olution galaxy spectrum which can be used to estimate mic variance (McDonald & Seljak 2009), and it improves redshift. These techniques are limited to an accuracy of photo-z determinations (Newman 2008; Zhang et al. 2010) σz = δz(1 + z) 0.07(1 + z) for a ground-based survey or Our calculations below and (Gazta˜naga et al. 2012) show ≈ σz 0.05(1+z) for a space-based survey each using 5 filter that DESI-like spectroscopic redshifts combined with a high ≈ ∼ bands. The benefit is that they are significantly faster than quality imaging survey boosts our ability to measure the equivalent spec-z surveys, capturing a couple of orders of DETF Figure of Merit. This improvement is stronger when magnitude more galaxies per observing night (Hildebrandt the spectroscopy and imaging overlap . Tests of General Rel- 2012). ativity benefit even more from the combinations of RSD and A new generation of high-resolution photometric sur- weak lensing because each responds differently to combina- veys, such as PAU (Ben´ıtez 2009a), are also planned. These tions of the two metric potentials. Again, having same sky aim to fill a gap between the standard spec-z and photo-z configuration gains an additional improvement. However, we surveys by using up to 50 filter bands to achieve photometric note that other calculations (e.g. BigBOSS White Paper, redshift reconstructions of much greater accuracy. Surveys Cai & Bernstein (2012); de Putter et al. (2013)) do not find of all these types are major investments in terms of money, improvement from same sky. The source of the discrepancy instrument time and staff-time with observing time alone maybe due to the implementation of the covariance matrix counted in hundreds of nights. calculations, assumptions about galaxy biasing for LSS and In general, the quality of imaging surveys is essential for intrinsic alignment for WGL, the range of k-values, and the the success of a multi-object spectroscopic survey on differ- assumed sky area and the redshift distribution of the spec- ent levels, at increasing demand on image quality: (i) imag- troscopic sample. ing is critically required to create a catalogues of objects for fibre allocation; (ii) the photometric quality and number of filters impact the success rate of selection of LRGs, ELGs, 3 A UNIFIED C(L)S FRAMEWORK and z > 2.1 QSOs; (iii) the images can be used for shape measurements for weak lensing (cosmic shear) and hence In this paper we have made a decision to describe all our enhance the science as described below; (iv) many other sci- observables, for both the spec-z and photo-z surveys, us- ence byproducts may result from combining imaging and spectroscopy, e.g. for detailed studies of galaxy evolution. Combining imaging and spectroscopy could be useful for 1 http://www.astronomy.ohio-state.edu/ dhw/jwg.pdf
c 2009 RAS, MNRAS 000, 1–21 4 Donnacha Kirk, Ofer Lahav, Sarah Bridle, Stephanie Jouvel, Filipe B. Abdalla, Joshua A. Frieman
Instrument Telescope No. Galaxies Sq. Deg.
SDSS I + II APO 2.5m 85K LRG 7,600
Wiggle-Z AAT 3.9m 239K 1,000
BOSS APO 2.5m 1.4M LRG + 160K Ly-α 10,000
HETDEX HET 9.2m 1M 420
eBOSS APO 2.5m 600K LRG + 70K Ly-α 7,000
DESI NOAO 4m 32M LRG + 2M Ly-α 18,000
SUMIRE PFS Subaru 8.2m 4M 1,400
4MOST VISTA 4.1m 6-20M bright objects 15,000
EUCLID 1.2m space 75M 14,700
Table 1. Summary of current or planned BAO capable spectroscopic surveys. Based on table 4 of the MS-DESI Science Alternatives Report. (Parkinson 2012; Schlegel et al. 2009; Comparat 2013a; Hill 2008; Sugai 2012b; de Jong 2012; Amiaux 2012; Abdalla 2012; Pilachowski 2012),http://www.sdss.org
Instrument Telescope Observing Bands No. Galaxies Sq. Deg.
DES Blanco 4m g,r,i,z,y 300M 5000
KiDS VST2.6m u,g,r,i 90M 1500
VHS Vista4m Y,J,H,Ks 400M 20,000
Viking Vista4m Z,Y,J,H,K - 1,500
HSC Subaru 8.2m g,r,i,z,y 400M 2,000
Pan-STARRS 1 Hawaii 1.8m g,r,i,z,y 1B 30,000
PAU WHT 4m 40 narrow-band 30,000 100-200
J-PAS OAJ 2.5m 54 narrow-band 14M LRG 8,000
Skymapper SSO 1.35m u,v,g,r,i,z - 20,000
LSST LSST 8.4m u,g,r,i,z,y 4B 20,000
Euclid 1.2m space R + I + Z,Y,J,H 1.5B 14,700
Table 2. Summary of current or planned photometric surveys for LSS and/or WGL. (Soares-Santos & DES Collaboration 2012; de Jong et al. 2013; Fleuren 2012; Ben´ıtez 2009b; Taylor 2013; Keller et al. 2007; Amiaux 2012; Ivezic 2008) http://www.ast.cam.ac.uk/∼rgm/vhs/, http://www.naoj.org/Projects/HSC/, http://ps1sc.org/
i ing projected angular power spectra, C(l)s. This enables where WX (l,k) is the window function for observable X, to- us to use the same formalism for cosmic shear, LSS and mographic bin i. We describe the window functions used in RSDs. More importantly it provides a language in which this paper in section 3.1 below. P (k) is the nonlinear mat- the cross-correlation between probes, their joint data vec- ter power spectrum today, k denotes wavenumber, measured tor and joint covariance can be written without resort to in hMpc−1, and l denotes angular multipole (Fisher et al. any special machinery or complicated, untested derivations. 1994). In addition we can include systematic effects in a consistent We consider cosmic shear, denoted by ǫ, and galaxy way for all probes. This paper treats galaxy bias, the galaxy- clustering, denoted by n. This gives us three different ij shear correlation coefficient and photometric redshift error. C(l) observables: Cnn(l), the galaxy-galaxy correlation (else- ij It is straightforward to expand to Intrinsic Alignments (IAs) where called galaxy clustering), Cǫǫ (l), the shear-shear cor- ij (Joachimi & Bridle 2010) and other systematics. relation from WGL and Cnǫ(l), the galaxy-shear cross- A general C(l) is the projection of two window func- correlation. tions where each corresponds to the projection kernel of a We assume that we have access to two surveys: an op- particular observable for a particular tomographic bin. The tical survey of 300 million galaxies with photometric qual- projected angular power spectrum for observable X in bin i ity redshifts and sufficient resolution to perform shape mea- and observable Y in bin j is given by surement for WGL, and a survey of 10 million galaxies
i,j 2 i j 2 with spectroscopic quality redshifts. We will make use of C (l)= WX (l,k)W (l,k)k P (k)dk, (1) XY π Z Y these surveys throughout the paper and refer to them as our
c 2009 RAS, MNRAS 000, 1–21 Optimising Spectroscopic & Photometric Galaxy Surveys: Same-sky Benefits for Dark Energy & Modified Gravity 5
“photo-z survey” and our “spec-z survey”. We split each sur- The distortion caused by coherent infall velovities takes vey into a number of tomographic bins in redshift. The spec- a particularly simple form in Fourier space, given by the z quality redshift allows us to bin these galaxies at much familiar Kaiser formula (Kaiser 1987) higher resolution in z. We choose 5 tomographic bins for s 2 the photo-z survey (consistent with DES-like surveys in the δg(k, µ)=(bg(k,z)+ f(z)µ )δm(k) (5) literature) and 40 tomographic bins for our spec-z survey where µ is the cosine of the angle between k and the line- (still computationally feasible and giving sufficient redshift of-sight, the superscript s denotes redshift-space, bg(k,z) is resolution to capture the bulk of the available information the galaxy bias and and f(z) = dlnD(z) Ω0.55 (Peebles dlna ≈ m (Asorey et al. 2012)). For simplicity we assume our cosmic 1980). shear observable, ǫ is always from the photo-z survey while We extend the galaxy window function (eqn.2) to in- our galaxy clustering observable, n, comes from the spec- clude the effects of RSDs following Fisher et al. (1994); z survey. The cross-correlation observable nǫ, where it is Heavens & Taylor (1995); Padmanabhan et al. (2005) where present, uses galaxies from both surveys. More details on they express the RSDs as an additional term in the galaxy our fiducial survey assumptions are given in section 4. clustering window function. The total LSS weight function is given by 3.1 Weight Functions i i i The projected angular power spectrum for a particular probe Wn,tot(l,k)= Wn(l,k)+ Wn,R(l,k). (6) and combination of tomographic bins is obtained by includ- That is, the sum of the non-RSD weight function (eqn.2) ing the appropriate weight functions in the general C(l)s and a new RSD term equation, given in eqn. 1, above. The probes we consider are 2 cosmic shear, ǫ, and galaxy clustering, n. i (2l + 2l 1) Wn,R(l,k)= β f(y) − jl(ky) The weight function for galaxy clustering is Z (2l + 3)(2l 1) h − i i l(l 1) (l + 1)(l + 2) − jl−2(ky) jl+2(ky) dy. Wn(l,k)= bg(k,z)n (z)jl(kχ(z))D(z)dz, (2) − (2l 1)(2l + 1) − (2l + 1)(2l + 3) Z − i i (7) where bg (k,z) is the galaxy bias, n (z) is the galaxy redshift distribution of tomographic bin i, D(z) is the linear growth This approach does not model the ‘Finger of God’ function and jl(kχ(z)) is the l-th order speherical Bessel effect- small scale RSDs due to the virial motion of galaxies function of the first type (Huterer et al. 2001). within clusters (Kang et al. 2002). As we cut our galaxy ob- The weight function for cosmic shear is servables to exclude non-linear scales (see section 4.9 below) we feel justified in ignoring this effect in our model. The i i Wǫ (l,k)= q (z)jl(kχ(z))D(z)dz, (3) lensing kernel is broad enough to wash out any effects from Z RSDs so we leave eqn. 3 unchanged and ignore RSDs for our i where q (z) is the lensing weight function, given by WGL observables. 2 χ ′ 3H Ωm χ(z) ′ ′ χ(z ) χ(z) qi(z)= 0 dχ ni χ(z ) − , (4) 2c2 a(z) Z χ(z′) χhor 4 FORECASTING ASSUMPTIONS where χ is comoving distance, ni(χ) is the galaxy red- shift distribution of tomographic bin i (Takada & Jain 2004; Here we detail the fiducial assumptions we make when fore- Joachimi & Bridle 2010). casting the science results of our photo-z and spec-z surveys. Each of these weight functions are constructed for a We begin with a summary of the Fisher Matrix (FM) for- particular tomographic redshift bin, i, defined by the galaxy malism which we use to make our forecasts and the way in i redshift distribution, n (z). Together we can use these which this, combined with our C(l)s approach, lends itself weight functions to define three probes based on 2-point readily to the combination of probes from different surveys. functions: the shear-shear correlation, ǫǫ, the galaxy-galaxy Next we describe the survey strategy and target selec- correlation, nn, and the galaxy-shear cross-correlation, nǫ. tion assumptions we make for both surveys. These are gen- erally held fixed in this paper but are explored in detail for the spec-z survey in our companion paper (Jouvel 2013). 3.2 Redshift Space Distortions The C(l)s formalism described above is general for When considering a survey with sufficiently high resolution cosmic shear and galaxy clustering (including RSDs) in redshift information it is possible to learn more about LSS a ΛCDM cosmology. Here we describe in detail the as- than the galaxy positions alone provide. Galaxies, as well sumptions we make on elements of the formalism including as moving as part of the underlying Hubble flow, have their galaxy bias, the galaxy-shear cross-correlation coefficient, own peculiar velocities, sourced by local gravitational po- non-linear clustering and the range of scales considered. We tentials, past mergers etc. In a galaxy redshift survey a net also describe an extension to the formalism which describes peculiar velocity along the line of sight away from (towards) deviations from General Relativity (GR). It is this exten- the observer adds to (subtracts from) the apparent redshift sion which allows us to forecast the ability of our probes to of a given galaxy. The impact of these effects on the survey constrain deviations from GR. Several of these assumptions are known as Redshift Space Distortions (RSDs) and can be are subsequently varied in section 5 where we study their used to learn about cosmology as they are sourced by the impact on the constraining power of the individual surveys local gravitational potential. and their joint combinations.
c 2009 RAS, MNRAS 000, 1–21 6 Donnacha Kirk, Ofer Lahav, Sarah Bridle, Stephanie Jouvel, Filipe B. Abdalla, Joshua A. Frieman
4.1 Fisher Matrices If the only observables being considered are ǫǫ or nn then only the covariance sub-matrices Covijkl(l) and We make our forecats under the Fisher Matrix formalism. ǫǫǫǫ Covijkl (l) respectively need be considered. To forecast constraints on cosmological parameters we calcu- nnnn We recognise that the projected angular power spectra late the Fisher information matrix (see e.g.Heavens (2009)), formalism has some limitations. Even with a large number which is the expectation value of the Hessian matrix of the of tomographic bins, there is still a loss of information due to log likelihood with respect to some parameters α, β, projection along the line of sight within bins which, after all, ∂2lnL have some finite width. While several analyses have shown Fαβ Hαβ = (8) ≡h i − ∂θα∂θβ that this formalism obtains all the available cosmological and can be written information from a photo-z WGL survey (Joachimi & Bridle 2010; Kirk et al. 2011), it is likely that some information is lmax i,j mn ∂D (l) −1 ij mn ∂D (l) lost in our analysis of the spec-z LSS survey compared to Fαβ = Cov D (l), D (l) , ∂pα ∂pβ a full 3D analysis. Nevertheless, we feel justified in using a l=Xlmin (i,j)X,(m,n) h i (9) C(l)s approach as Asorey et al. (2012) have shown it to be where D(l) is the data vector under consideration, competitive even for spec-z surveys and it provides many benefits in terms of joint systematic and covariance matrix Cov [Dij(l), Dmn(l)] is the covariance matrix, pα label the cosmological and nuisance parameters we vary in estimation not available to a mixed C(l)/P (k) approach. our analysis and i, j label pairs of tomographic bins (Joachimi & Bridle 2010). The FM formalism provides an 4.3 Cosmological Parameters estimate of the the marginalised error on each cosmolog- ical parameter through the Cramer-Rao inequality, σi > We assume a fiducial set of late-Universe cosmolog- −1 (F )αα. This is the marginalised parameter error, the ical parameters, pα = Ωm,w0,wa,h,σ8, Ωb, ns = { } independentp error on parameter α, i.e. the error if all other 0.25, 1, 0, 0.7, 0.8, 0.05, 1 , where Ωm & Ωb are the di- { − } parameters are fixed, is given by σi > 1/Fαα. mensionless matter and baryon densities respectively (i.e. p Ωcdm = Ωm Ωb), w0 and wa parameterise the DE equation − 4.2 Combined Probes of state, w(z)= w0 +waz/(1+z), h is the Hubble parameter, σ8 is the normalisation of the matter power spectrum, ns is The above formalism shows how cross-correlations between the slope of the primordial power spectrum and δz is the observables come naturally in the C(l)s formalism. In our error on the photometric galaxy redshift distribution. We case we can trivially construct a data vector including both also use a number of nuisance parameters for galaxy bias, galaxy and shear correlations and their cross-correlations, bX , described in section 4.6 below. We allow a single global i,j ij ij ij D (l) = C (l),C (l),C (l) . This models the avail- photo-z error nuisance parameter, δz, described in section { ǫǫ nǫ nn } able cosmological information when our two surveys over- 4.8. All quoted results are marginalised over this parameter lap on the sky. It is also a simple matter to calculate the space. We assume flatness throughout as a theoretical prior. full covariance matrix between observables, including all the We apply wide, flat, uninformative priors to all cosmological off-diagonal elements (Joachimi & Bridle 2010; Bernstein and nuisance parameters. We also, where appropriate, as- 2009), sume two modified gravity (MG) parameters Q0 & R0, a set ijkl ijkl ijkl of nuisance parameters for the galaxy-shear cross-correlation Covǫǫǫǫ(l) Covǫǫnǫ(l) Covǫǫnn(l) ijkl ijkl ijkl coefficient, rg, and an extended set of photo-z nuisance pa- Cov(l)= Covnǫǫǫ(l) Covnǫnǫ(l) Covnǫnn(l) rameters. These are detailed in sections 4.7, 5.3.4 and 5.4 Covijkl (l) Covijkl (l) Covijkl (l) nnǫǫ nnnǫ nnnn below. (10) We calculate each individual covariance sub-matrix as (ijkl) (ij) (kl) ′ Covαβγδ (l) ∆Cαβ (l)∆Cγδ (l ) 4.4 Figures of Merit ≡ D E 2π (ik) (jl) (il) (jk) It is often convenient to summarise the cosmological con- ′ ¯ ¯ ¯ ¯ = δll Cαγ (l)Cβδ (l)+ Cαδ (l)Cβγ (l) Al∆l n o straining power of a survey or combination of surveys in one (11) number. This is of course a great simplification which pays where C¯ accounts for shot and shape noise as: no attention to many potential benefits of a particular sur- vey design but it has the benefit of allowing easy comparison C¯(ij)(l) C(ij)(l)+ N (ij) (12) αβ ≡ αβ αβ between survey designs, assumptions and even the results of and the noise contributions are given by different papers. 2 Throughout this paper we will quote values for the Dark (ij) σǫ Energy Figure of Merit (DE FoM), based on the Dark En- Nǫǫ = δij (i) (13) 2¯ng ergy Task Force (Albrecht et al. 2006) definition,
(ij) 1 Nnn = δij (14) 1 (i) FoMDE = , (16) n¯g −1 4 det(F )DE N (ij) = 0 (15) p nǫ where the subscript DE denotes the 2 2 sub-matrix of the 2 × where σǫ is the shape noise from cosmic shear galaxy shape inverse FM that corresponds to the entries for the equation measurement andn ¯g is the shot noise due to the fact we of state of DE parameters, w0 and wa. Note that differ- observe a finite number of galaxies. ent prefactors to this equation exist in the literature. We
c 2009 RAS, MNRAS 000, 1–21 Optimising Spectroscopic & Photometric Galaxy Surveys: Same-sky Benefits for Dark Energy & Modified Gravity 7 use the factor of 1/4 for consistency with related papers Parameter Photo-z WGL Spec-z LSS (Bridle & King 2007; Joachimi & Bridle 2010). By analogy we define a Modified Gravity Figure of Area [deg2] 5000 5000 Merit (MG FoM), zmin 0 0 zmaz 3 1.7 1 FoMMG = , (17) Nz 5 40 −1 4 det(F )MG δ(z) 0.07(1 + z) 0.001(1 + z) n [arcmin−2] 10 0.56 where the subscript MG denotesp the 2 2 sub-matrix of the g × inverse FM that corresponds to the entries for the equation Q0(1+R0) Table 3. Fiducial survey assumptions for our photo-z WGL sur- of state of DE parameters, Q0 and R0 (Kirk et al. 2 vey and spec-z LSS survey. Nz is the number of tomographic 2011). Our MG parameters are held fixed when the DE FoM redshift bins of equal number density. δz is the gaussian redshift is calculated but the DE parameters are allowed to vary error and ng is the number density of (usable) targets of the sky. when we calculate the MG FoM because our survey com- More details on fiducial survey assumptions can be found in sec- bination needs to be able to constrain expansion history tion 4.5. as well as deviations from GR (from mis-matched expan- sion/growth of structure). density over this range because we do not want our results to depend on a particular target selection strategy. We as- 4.5 Survey Strategy sume high quality redshift information, δz = 0.001(1 + z), which we use to split the survey into 40 tomographic bins. We assume two fiducial surveys: a photo-z WGL survey and See our companion paper Jouvel (2013) for a detailed anal- a spec-z LSS survey including RSDs. Each is modelled gener- ysis of how target selection and survey design effects this ically so that our results are as widely applicable as possible. spec-z survey. In this paper we explore systematic effects Particularly in the spec-z case our toy n(z) does not look and theoretical assumptions. like the true redshift distribution that any particular survey Our fiducial survey assumptions are summarised in ta- would measure but it has the benefits of simplicity, clarity ble 3. in this paper we study the impact of survey area and generality. We can examine the impact of forecast as- and photo-z/spec-z overlap on constraining power but leave sumptions without dealing with the complicated interplay other properties fixed for clarity. between z-coverage, nuisance parameters, ell-cuts etc. and a particular, feature-full n(z). We examine the impact of a specific survey target selection in section 5.1 and the whole 4.6 Galaxy Bias issue is investigated in detail as part of our companion paper Jouvel (2013). As well as our standard cosmological parameters we must We model our photo-z survey on the Dark Energy Sur- also consider a number of ‘nuisance parameters’ which de- vey (DES) which saw first light in September 2012 and scribe systematic effects for which we lack a physical model is due to start full survey operations in September 2013. or the uncertainties on our model contribute significantly We assume 300 million galaxies are observed over 5000 to the overall error budget of our experiment. In this work 2 deg with a gaussian photometric redshift error of δz = the principle systematic effects are the galaxy bias, bg, and 0.07(1 + z). The survey covers a redshift range of 0 c 2009 RAS, MNRAS 000, 1–21 8 Donnacha Kirk, Ofer Lahav, Sarah Bridle, Stephanie Jouvel, Filipe B. Abdalla, Joshua A. Frieman which is designed to be agnostic about the form of bg but metric potentials, to include enough uncertainty in both scale and redshift de- Φ pendence that we do not over-constrain cosmology due to R = . (24) Ψ naive assumptions about our knowledge of bias. Our model follows that of Joachimi & Bridle (2010). Both Q and R are unity in GR. In general MG theories both This fiducial model takes the form could vary as a function of scale and redshift. In this work we assumed, for simplicity, that both are scale independent bg(k,z)= Abg Qbg (k,z), (20) (our analysis cuts off at scales much larger than those where a screening mechanism would need to be invoked to preserve where Abg is a variable amplitude parameter with fiducial solar system GR tests) and follow a simple evolution with value Abg = 1 and Qbg (k,z) is a free function in k,z formed s s redshift, Q = Q0a , R = R0a , with s = 3. This evolution by the interpolation of a grid of Nz Nk nodes in k,z, each × is motivated by the need to preserve CMB and BBN tests of which is allowed to vary independently and has a fiducial of GR at high redshift, allowing the modification to “turn value of unity. The Nz nodes are spaced linearly throughout on” at late times in an effort to explain cosmic acceleration the z-range of our survey while the Nk nodes are log spaced − without invoking Dark Energy. between k = 0.001 and 30hMpc 1. In total this formalism We fix s = 3 but allow Q0 and R0 to vary from their introduces 1+ Nz Nk nuisance parameters for galaxy bias. × fiducial GR values. When Q0 = 1 or R0 = 1 the devia- Our fiducial model assumes a 2 2 grid for galaxy bias, i.e. 6 6 × tion from GR enters our observables in two ways: through 5 nuisance parameters. We investigate the impact of this changes to the linear growth factor which effects all our ob- choice in section 5.3.3 below. servables ǫǫ, nǫ and nn as well as the β term in the RSD When we consider the galaxy-shear cross correlation, kernel and through changes to the geometric factor in our Cnǫ(l), there is another parameter that must be defined. cosmic shear kernel, adding a factor of Q(z)(1+R(z)) to the The galaxy-shear correlation coefficient, rg(k,z), which nǫ observable and a factor of [Q(z)(1 + R(z))]2 to ǫǫ. mediates between the galaxy and shear power spectra (Guzik & Seljak 2001; Mandelbaum et al. 2013), 4.8 Photometric Redshift Errors Pnǫ(k,z) rg(k,z)= . (21) Pǫǫ(k,z)Pnn(k,z) There has been some interest in the use of combined photo-z p and spec-z surveys to “self-calibrate” the photometric red- If there is uncertainty about the form of this parame- shift error of the photo-z survey (Newman 2008). In our fidu- ter it can contribute a substantial systematic error to cial set-up we assume a single global error for the photo-z the modelling of the shear-position, nǫ, cross-correlation. n(z), σz = δz(1 + z), where δz is a free parameter allowed Gazta˜naga et al. (2012) argue that, if analysis is restricted to vary around its fiducial value of 0.07. We keep this model to sufficiently large scales and galaxy bias is assumed to be simple to ensure that our combined results do not rely too scale-independent, then it is acceptable to set rg = 1. For heavily on this photo-z calibration effect. most results below we fix rg at unity but in section 5.3.4 We investigate the impact of this calibration in more we consider the impact of this assumption when we com- detail in section 5.4 below. In that section we explore joint pare results for rg = 1 and rg parameterised as rg(k,z) = constraints in the case of the photo-z error being accurately Arg Qrg (k,z), similarly to the fiducial galaxy bias param- known and in the more sophisticated case where we allow a eterisation above. In this case another 1 + Nz Nk nui- i × variable δz for each tomographic bin i in the photo-z survey. sance parameters are introduced to parameterise the cross- In addition we allow the mean redshift of the photo-z tomo- correlation. The aim of introducing extra nuisance parame- i graphic bins to vary via parameters ∆z, called photometric ters is to avoid a systematic biasing of our constraints from redshift bias for bin i, with fiducial values zero. This ap- poor modelling of rg, at the cost of reduced statistical error. proach gives us a very flexible photo-z error, we can reduce the uncertainty by applying increasingly tight priors to the photo-z nuisance parameters. 4.7 Modified Gravity We also present results forecasting the ability of our sur- 4.9 Non-Linear Matter Clustering veys to constrain deviations from gravity as described by General Relativity (GR). To describe these Modified As matter collapses under gravity, higher density regions Gravity (MG) scenarios, we introduce two new parame- reach a point at which their local overdensity is of order ters, Q, R, following the formalism of Laszlo et al. (2011); unity. In this regime structure formation no longer continues Bean & Tangmatitham (2010) This formalism treats pertur- under the well behaved linear growth equations but contin- bations to the metric in the Newtonian gauge, ues to collapse in a much more complicated non-linear man- 2 2 2 2 2 i j ner. For our forecasts we produce a linear matter power spec- ds = a (τ)[(1 + 2Ψ)] dτ + a (τ) (1 2Φ)a ) δij dx dx − − trum using the fitting function of Eisenstein & Hu (1998) (22) and model the non-linear growth of structure using the Q describes changes to the Poisson equation which describes halofit model of Smith et al. (2003). However, it is known how the Newtonian potential is sourced by matter, that this non-linear fitting is only accurate to 10% so we ∼ k2Φ= 4πGQaaρ(a)δ, (23) are not overly confident of our ability to forecast non-linear − growth or, even more so, the non-linear galaxy bias and and can be considered as an effective Newton’s constant. RSDs at small scales. R0 parameterises the ratio of the Newtonian & Curvature If naively included the non-linear scales contribute a c 2009 RAS, MNRAS 000, 1–21 Optimising Spectroscopic & Photometric Galaxy Surveys: Same-sky Benefits for Dark Energy & Modified Gravity 9 800 from combining them? Fig 2 shows 95% confidence contours for DE & MG respectively, assuming our fiducial survey sce- 700 Fiducial narios. In the case of the DE constraints, GR is assumed. Rassat et al. 2008, X=0.2 The MG constraints are marginalised over w0,wa. All other 600 cosmological parameters are marginalised over in both anal- yses, as are a 2 2 grid if bg parameters. rg = 1 is assumed. × Both marginalise over our standard set of cosmological pa- 500 rameters, the galaxy bias nuisance parameters and a single photo-z error parameter. 400 max l Constraints are shown for the photometric WGL sur- vey alone, ǫǫ, the spectroscopic LSS survey alone , nn, the 300 independent combination of the two surveys (this is the constraint from two surveys on separate areas of the sky), 200 ǫǫ + nn, and the dependent combination of both surveys including all cross-correlations (which includes the full in- 100 formation for the case where the survey areas fully overlap), ǫǫ + nǫ + nn. 0 The DE and MG constraints in Fig 2 share a num- 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 z ber of features. In each case the photometric and spectro- med scopic surveys alone are relatively poorly constraining. This is unsurprising given the cosmology & nuisance parame- ters marginalised over and our extremely conservative k-cut Figure 1. The fiducial l-cuts used in this work based on the to remove non-linear scales where bg and matter clustering recipes of Joachimi & Bridle (2010) [blue solid] and Rassat et al. generally are very uncertain. However, even given this lim- (2008) with X=0.2 [red dashed]. The plot shows the maximum l itation, the combination of the spectroscopic nn with the value included in the analysis as a function of tomographic bin photometric ǫǫ is extremely beneficial. Adding the surveys number, N . In the case of the nǫ correlation, the n bin defines bin (ǫǫ + nn) produces a factor of 4.5 improvement in DE the cut. In the case of the nn correlation we chose the optimistic ∼ FoM compared to the photo-z only survey, without includ- case and cut on the higher redshift bin. lmin = 10 is assumed throughout. For more details see section 4.9 ing cross-correlations. When these cross-correlations are in- cluded we see a ‘same-sky benefit’ of almost a factor of four in DE FoM. This improvement comes from the inclusion great deal of constraining power to our LSS observables. of the nǫ cross-correlation observable which introduces new Rather than over-constrain cosmology and galaxy bias by cosmological dependencies and breaks some degeneracies be- including effects which we understand so poorly, we cut non- tween cosmological parameters and galaxy bias. linear scales from our analysis wherever the n observable In the MG panel of Fig. 2 the nn only survey is slightly more constraining than ǫǫ alone. This is due to the ability appears, i.e. in Cnn(l) and Cnǫ(l). We follow the approach of the high-resolution RSD measurements to probe varia- of Rassat et al. (2008) by defining some kmax which we con- tion in the linear growth factor. There is clearly a strong vert into an lmax(z) which defines a maximum multipole degeneracy between Q & R that the nn probe alone can- per tomographic bin. For consistency we choose kmax = 0 0 0.132 zi as used in Joachimi & Bridle (2010), where zi not overcome. It requires the extra geometric information ∗ med med is the median redshift of tomographic bin i. It should be contained in the WGL probe to break this degeneracy and noted that this is not exactly the same recipe as the fiducial produce a closed constraint on deviations from GR. This one set out in Rassat et al. (2008). We show the difference degeneracy-breaking means that the combination of ǫǫ + nn in Fig. 1. We are happy with our fiducial choice as it is the produces a much more pronounced improvement than in the more conservative of the two. We investigate the impact of DE case, giving more than a factor of 150 improvement over these k-cuts based on the Rassat et al. (2008) recipe and WGL alone. The inclusion of the nǫ cross-correlation with its give more quantitative details in section 5.8. Scales smaller own dependence on Q0, R0 adds another, same-sky, beneift than this are removed. Fig. 1 shows the cut in angular scale, of more than a factor of two. The same-sky benefit is less l, that we deploy as a function of redshift. For the nǫ cor- pronounced for MG than for DE because of the huge im- relation we cut based on the galaxy clustering tomographic provement gained even from the independent combination bin. The nn correlation offers a choice of tomographic bin of the two probes, making the cross-correlation improvement to cut on (in the case of cross-bin correlations), we choose proportionally less important. the optimistic option and cut on the higher redshift bin. The We note that the nn only probe gets nearly all its con- exact details of this cut are investigated in section 5.8 below. straining power for DE and MG from the inclusion of RSDs, without these the marginalistion over bg reduces the LSS- only constraints to near zero. In this case the full joint con- straints rely even more strongly on cross-correlation with 5 FORECASTS the WGL photo-z survey, producing a same-sky benefit fac- Given the forecasting assumptions detailed in the previous tor of 4.7 and a combined ǫǫ + nǫ + nn DE FoM nearly ∼ section, we want to answer the primary questions: how well equal to that of the case where RSDs are included. MG can our example photometric & spectroscopic surveys con- constraints suffer greatly from the loss of RSDs. The LSS strain cosmology? and what particular benefit do we gain probe is no longer able to provide an orthogonal constraint c 2009 RAS, MNRAS 000, 1–21 10 Donnacha Kirk, Ofer Lahav, Sarah Bridle, Stephanie Jouvel, Filipe B. Abdalla, Joshua A. Frieman 5 on deviations from GR and the final joint constraint with ee FoM = 1.6 cross-correlations is over an order of magnitude smaller in 4 nn FoM = 0.9 ee + nn FoM = 7.2 terms of MG FoM than when RSDs are included. ee + ne + nn FoM = 27.8 In the following sub-sections we perturb in turn a range 3 of the assumptions we have made for our fiducial forecast. 2 Some of the results are summarised in table 4 for ease of comparison. 1 a 0 w 5.1 Survey Specific Target Selection −1 We have assumed a generic survey strategy for our spec-z survey in order that our results might be as widely applicable −2 as possible. Another benefit of assuming a constant galaxy −3 density out to z = 1.7 is that our results are not hostage to the particularities of a certain target selection strategy −4 which could interact in a peculiar way with, for example, redshift coverage, photo-z error or k/z-dependence of galaxy −5 −2 −1.5 −1 −0.5 0 bias. w 0 We recognise that no individual survey as carried out 2 will look exactly like the toy survey we assume here. The ee MGFoM = 0.3 impact of specific survey and target selection strategies are 1.8 nn MGFoM = 0.6 considered in our companion paper Jouvel (2013). In this ee + nn MGFoM = 50.1 ee + ne + nn MGFoM = 112.7 section we reproduce the constraints of one representative 1.6 survey drawn from that paper to illustrate the differences 1.4 with respect to the survey model we assume here. We choose a 5000 deg2 survey with an exposure time of 20 minutes 2 1.2 using a 4000 fibre spectrograph with a 3 deg field of view. )/2 0 We assume two galaxy populations, LRGs and ELGs are 1 separately targeted in a ratio of 30/70. Assuming 8 hours (1+R 0 observing per night and a 10% overhead in survey time, we Q 0.8 estimate that it would take about 139 observing nights to 0.6 saturate the available target list. Despite the differences in survey strategy between our 0.4 toy model and this more specific example, the basic trends in DE & MG constraints from spec-z/photo-z combinations 0.2 are relatively robust. While the nn survey alone is less con- straining due to a more uneven z-distribution and slightly 0 −4 −2 0 2 4 6 smaller z-range, the independent ǫǫ+nn combination contin- Q 0 ues to improve on the WGL-alone constraint by more than a factor of three in the case of DE and nearly two orders of magnitude in the case of MG. The same-sky benefit from Figure 2. Upper panel: Marginalised 95% errors on w0, wa for including the nǫ correlations is more pronounced with these ǫǫ from our photo-z survey [green contours], nn from our spec-z specific survey assumptions. In fact the joint DE constrain- survey [black contours], their independent combination, ǫǫ + nn, ing power is better than for our fiducial scenario while the [red contours] & their combination including cross-correlations, joint MG result is lower but roughly similar. What is clear ǫǫ + nǫ + nn, [blue contours]. Our fiducial survey strategies are is that none of the simplifying assumptions we have made assumed. 5 tomographic bins are used for the photo-z survey, 40 in our fiducial scenario badly bias the trends we are inter- tomographic bins for the spec-z survey. We marginalise over our ested in exploring when our spec-z and photo-z surveys are standard cosmological parameters, assuming GR. We marginalise over galaxy bias, assuming one overall amplitude term and a 2×2 combined. Nevertheless we will continue to use the simple fiducial spec-z survey scenario so that the other assumptions grid of bg(k,z) nodes, while fixing rg(k,z) = 1. We marginalise over a single global photo-z error term. Any observable contain- we explore below can be quantified without a complicated ing galaxy clustering is assumed to be cut at large ell. RSDs are interplay with the irregular n(z) that is the result of some included for the spec-z survey. Lower panel: Marginalised 95% specific target selection assumptions. errors on Q0, Q0(1 + R0)/2 for ǫǫ from our photo-z survey [green contours], nn from our spec-z survey [black contours], their inde- pendent combination, ǫǫ+nn, [red contours] & their combination 5.2 CMB Prior including cross-correlations, ǫǫ + nǫ + nn, [blue contours]. We marginalise over w0 and wa, all other assumptions are the same This paper is primarily concerned with the details of the as for the upper panel. combination of a spec-z galaxy clustering/RSD survey with a photo-z WGL survey. It is from this sort of joint probes analysis that all the most stringent constraints on cosmol- ogy will be derived. Of course there are cosmological probes c 2009 RAS, MNRAS 000, 1–21 Optimising Spectroscopic & Photometric Galaxy Surveys: Same-sky Benefits for Dark Energy & Modified Gravity 11 Scenario Photo-z Spec-z Photo-z + Spec-z Photo-z × Spec-z Same-sky Benefit DE MG DE MG DE MG DE MG DE MG Fiducial 1.6 0.3 0.9 0.6 7.2 50.1 27.8 112.7 3.9 2.25 ‘Survey’ n(z) 1.6 0.3 0.2 0.1 5.4 24.7 55.9 93.1 10.4 3.8 With Planck 10.2 2.1 3.9 2.2 14.0 72.4 34.7 167.0 2.5 2.3 Marginalise rg 1.6 0.3 0.9 0.6 7.2 50.1 8.5 56.6 1.2 1.1 Fix bg 1.6 0.3 6.9 4.2 16.4 86.5 48.8 185.6 3.0 2.1 No RSDs 1.6 0.3 0.0 0.0 5.7 1.3 26.9 7.0 4.7 5.4 Fix δz 3.5 0.4 0.9 0.6 10.5 70.0 31.8 134.4 3.0 1.9 bg = 2 1.6 0.3 1.6 1.1 9.6 66.2 73.9 159.0 7.7 2.4 15,000 deg2 spec-z 1.6 0.3 2.7 12.6 11.3 201.9 38.6 423.4 3.4 2.1 Table 4. Summary of constraints from our photo-z and spec-z surveys for a number of forecast assumptions. We show DE & MG FoMs for Photo-z ǫǫ, Spec-z nn, Photo-z + Spec-z ǫǫ + nn and Photo-z × Spec-z ǫǫ + nǫ + nn, where the ǫ observables are always drawn from the photo-z survey and n are always drawn from the spec-z survey. We also show the Same-sky Benefit for DE & MG i.e. the FoM for ǫǫ + nǫ + nn, divided by that for ǫǫ + nn. We show the results for our fiducial surveys and fiducial assumptions and then perturb one assumption at a time: using a particular target selection/survey strategy scenario from Jouvel (2013); including Planck forecast priors; marginalising over rg the same way we do over bg; Fixing both bg and rg; removing RSDs from our n observables; fixing the photometric 2 redshift uncertainty, δz; changing the fiducial galaxy bias amplitude to bg = 2, increasing the area of the spec-z survey to 15,000 deg (with the same no.density per deg2) while keeping the area of the photo-z survey fixed at 5,000 deg2. Each change is made independently, all other assumptions are fixed at those of our fiducial forecasts. Details of these scenarios can be found throughout section 5. beyond LSS & WGL and any comprehensive constraints on cross-correlation produces a same-sky benefit factor of 2.5. ∼ cosmology will have to integrate them into the analysis. This is lower than the case without Planck but still a sub- stantial benefit. The MG same-sky benefit is robust to the When adding extra cosmological observables to our inclusion of Planck, unsurprising as the CMB can tell us analysis the most useful are those that provide orthogonal little about late-time modifications to GR. constraints on cosmological parameters, thereby breaking degeneracies present in the analysis and improving the fi- nal results. Adding “disjoint” cosmological probes which are 5.3 The Importance of Galaxy Bias sensitive to different physics is a powerful way of breaking degeneracies in particular observables. In our case LSS and The primary nuisance parameter we are interested in is WGL are both late-Universe probes sensitive to the growth galaxy bias, bg(k,z), which accounts for the fact that galax- of structure as the Universe evolves. Two of the most useful ies are a biased tracer of the underlying dark matter distri- probes to add to this mix are therefore type Ia supernovae bution. We have assumed a linear galaxy bias model, (SNe) and the Cosmic Microwave Background (CMB). The CMB probes the physics of the early Universe while SNe directly constrain the expansion history of the Universe (it δg = b(k,z)δm, (25) should be noted that WGL also has some direct access to where the galaxy overdensity, δg, is related to the matter expansion history through its geometric kernel). The best overdensity, δm, by a single function of scale and redshift. CMB observations to date come from the Planck satellite This propagates to a simple relation between the galaxy (Planck Collaboration et al. 2013). We have reproduced our power spectrum and the underlying DM power spectrum, fiducial results from Fig. 2 including a prior based on a forecast for the Planck mission [Dark Energy Survey The- 2 ory & Combined Probes group, private communication]. It Pgg(k,z)= b (k,z)Pmm(k,z). (26) would have been more complete to include the newly re- Our default parameterisation of this function allows a leased Planck results themselves in our combined data but single overall amplitude and variation in k/z-space through would have delayed the release of this paper. We are in- modulation of 2 2 grid nodes covering our full redshift formed [Tom Kitching, private communication] that the re- × range and all linear/quasi-linear scales. Galaxy bias is not leased Planck constraints and forecasts are similar enough particularly well understood or empirically constrained at for our present needs. present, especially on non-linear or quasi-linear scales (et al. Clearly the addition of the Planck constraints means 2013; Contreras 2013; Comparat 2013b; Pujol & Gazta˜naga that each of our previous probe combinations is correspond- 2013). It is this which motivates our stringent cuts in ell ingly more powerful than they were on their own, see Table (see section 5.8 for more details). Even so, the assumptions 4 for details. The strongest improvements come for the in- we make on galaxy bias, even at linear scales, can dramati- dividual probes with, for the case of DE, ǫǫ improving by cally effect the constraining power of our spec-z survey and a factor of 6 and nn by a factor of more than 4. What is the usefulness of same-sky correlations. These effects will be striking is how the combination of the WGL & LSS surveys, investigate din more detail in Clerkin et al. (in prep). with and without cross-correlation, still offer significant im- In this section we explore the impact of these assump- provements through the breaking of parameter degeneracies. tions on our constraints from the spec-z LSS survey and its While the independent, ǫǫ+nn+CMB, combination is only combination with our photo-z WGL survey. We divide our 40%˜ more powerful than ǫǫ + CMB, the addition of the nǫ assumptions about galaxy bias into three: the underlying c 2009 RAS, MNRAS 000, 1–21 12 Donnacha Kirk, Ofer Lahav, Sarah Bridle, Stephanie Jouvel, Filipe B. Abdalla, Joshua A. Frieman 5 rameterised by our 1 + nk nz nuisance parameters (grid × ee FoM = 1.6 nodes in k & z plus an overall amplitude term). In effect we 4 nn FoM = 0.2 assume that galaxy bias is constant with scale and redshift ee + nn FoM = 5.4 but we allow enough uncertainty to encompass the true b 3 ee + ne + nn FoM = 55.9 g evolution. Here we consider whether the choice of a fiducial 2 model can change our forecasts. As well as our fiducial bg = 1 model we also run fore- 1 casts assuming galaxy bias is given by the Fry (Fry 1996) and Q (Cole 2005) models. These are two regularly used galaxy a 0 w bias models, motivated by simple physical arguments and −1 N-body simulations. The Fry model gives a z-dependent bg, while the Q-model produces a k-dependent bg. −2 The Fry biasing model assumes the continuity equation and and linear growth, −3 b0 1 bg,Fry(z)=1+ − , (27) −4 D(z) −5 where D(z) is the linear growth function and b0 is known as −2 −1.5 −1 −0.5 0 w the “Fry parameter”, for which we assume a fiducial value 0 b0 = 2. 2 The Q model is motivated by a need to allow an un- ee MGFoM = 0.3 known scale dependence to enter the galaxy bias formal- 1.8 nn MGFoM = 0.1 ee + nn MGFoM = 24.7 ism. The Q-model, derived for low-redshift, once calibrated 1.6 ee + ne + nn MGFoM = 93.1 against N-body simulations populated by a semi-analytic galaxy formation model, takes the form 1.4 1+ Qk2 bg,Q(k)= , (28) 1.2 r 1 + 1.4k )/2 0 1 where we set Q = 4 following the usage in Swanson et al. (1+R (2010). 0 Q 0.8 We compare results from these different galaxy bias models in the unrealistic case where we assume that our 0.6 model exactly captures the physics of galaxy bias with 0.4 zero uncertainty. In this case, chosen for maximum differ- ence between models, there is significant scatter between 0.2 our results, with the Fry model in particular producing marginalised errors on w0,wa which are less than half those 0 −4 −2 0 2 4 6 of the fiducial model for some parameters. This can be un- Q 0 derstood because of the extra cosmology dependence en- coded in the Fry model, allowing an extra handle on the cosmology we’re trying to measure. The Q-model produces results in relatively good agree- Figure 3. Same forecast assumptions and layout as Fig 2 but ment with the fiducial model but this is an artifact of the here we assume a specific target selection and survey strategy for the spec-z survey. We assume a 5000 deg2 survey with 300 ob- stringent ell-cuts we apply to our LSS probes. As the signa- serving nights, 20min exposure time and a 30/70 split between ture features of the k-dependent (z-independent) Q-model LRG/ELG targets. All other assumptions are consistent with our only kick in for scales smaller than those we include, their fiducial spec-z survey. The WGL observables come from our fidu- effect is excluded from our forecast. if we cease to apply cial photo-z model survey. For more details see Jouvel (2013). cuts at quasi-linear scales, trusting our knowledge of non- linear physics up to l = 3000 (wildly optimistic) then we see large divergence between forecasts assuming our fiducial bg and those of the Q-model. In this regime the Q-model model of bg which we assume; the fiducial amplitude of bg over-constrains cosmology due to the non-physical nature of for the galaxy population captured by our survey and the its predictions for very small scales. level of uncertainty we allow to enter our bg model in the The divergence between forecasts for different fiducial form of nuisance parameters. We finish the section by ex- galaxy bias models reduces as the number of nuisance pa- ploring the impact of a related quantity, r , the galaxy-shear g rameters increases. cross-correlation coefficient. Galaxy bias model choice is an important and involved topic which we only have the space to scratch the surface of in this section. We have demonstrated the need to pay 5.3.1 Galaxy Bias Model attention to bg modelling when quoting results of LSS or For our fiducial model we assume that galaxy bias varies combined surveys. We are content that a sufficient number around unity in some redshift- and scale-dependent way pa- of nuisance parameters can dilute the difference between c 2009 RAS, MNRAS 000, 1–21 Optimising Spectroscopic & Photometric Galaxy Surveys: Same-sky Benefits for Dark Energy & Modified Gravity 13 models and that our fiducial nuisance parameterisation is tensive without significantly affecting the cosmological con- sufficient in this respect. straints produced. In particular we tested the z-dependent bias, bg(z), out to 40 nuisance parameters (one for each to- mographic bin) and found negligible decrease in constraining 5.3.2 Galaxy Bias Amplitude power as compared to the nz = 10 case. The photo-z WGL survey is insensitive to galaxy Recently there has been some attention in the literature to bias. When it is included, either with or without cross- the use of multiple, differently biased galaxy populations correlations, we not only see an overall improvement in FoM which can be used simultaneously as tracers of LSS in such a but there is a “floor” below which the FoM does not fall with way as to reduce cosmic variance (McDonald & Seljak 2009; increased grid flexibility, this can be thought of as the resid- Bernstein & Cai 2011). A full implementation of this ap- ual constraining power of the combination after nn has been proach would consider a number of differently biased galaxy marginalised out of existence. It is interesting that this floor populations and calculate C(l)s for their auto- and cross- is lower for the bg (k,z) case than for the bg(k) case, wtith correlations in all the available tomographic bin pairs. Each bg(z) lying only slightly above the bg(k,z) line. We expect galaxy population would require its own set of nuisance bg(k) to perform best in combination because it retains most parameters which could be “self-calibrated” through cross- information for nn-alone. correlation with the photo-z survey. The bottom left panel shows results in the case of the While such a full implementation of this approach is full same-sky combination ǫǫ+nǫ+nn. That each of the bias beyond the scope of this paper we want to underline the cases shows relative stability in the face of increased model importance of bias amplitude through a simple example. flexibility demonstrates the power of the cross-correlations We perform our fiducial forecast again but assuming to control unknown bias terms. The trend with increased that we have targeted a different, more strongly biased pop- number of grid nodes is less smooth than in the other cases. ulation of galaxies, setting bg = 2 and assuming the stan- This is not particularly unexpected- as the number of grid dard five nuisance parameters to allow uncertainty around nodes is changed, so is their spacing in k/z so we would not this new amplitude. require the FoMs produced to change monotonically. Never- As expected the spec-z survey alone is more constrain- theless this result does suggest that the exact location of our ing, by a factor of 1.8 in DE FoM. Interestingly while the b(k,z) “flexibility” can influence our results and should be independent non-overalpping combination of photo-z and treated with care. Our fiducial choice is conservative enough spec-z surveys is slightly improved (a factor of 1.3), the that we are not over-estimating our understanding of galaxy same-sky combination sees strong improvement by a factor bias, we are into the regime where the survey is ‘calibrating of 2.6 in DE FoM compared to the bg = 2 case. This means itself’, increasing the number of nuisance parameters would that the same-sky benefit factor goes from 3.9 with bg = 1 not affect our results overly. to 7.7 with b = 2 all thanks to the greater signal-to-noise g The same-sky benefit results (bottom right panel) cor- that comes from targeting more highly biased tracers. respond to the trend in the other plots. Same-sky bene- The trends in the MG case are similar but markedly fit improves with increased uncertainty in bg as the cross- less pronounced, with a change in same-sky benefit factor correlations act to calibrate the galaxy bias. The trend is from 2.25 with bg = 1 tot 2.4 with bg = 2. least pronounced in the case of bg(k) where the impact of increased bias uncertainty is limited. It should be noted that even in the case of a fixed bg (i.e. the un-justified assump- 5.3.3 Galaxy Bias Nuisance Parameterisation tion that we understand the bias term perfectly) there is still significant, 3, improvement from same-sky, confirming Fig. 4 shows DE FoM as a function of the number of grid ∼ × nodes marginalised over in bg(k,z), showing spearate re- that the cross-correlation’s effects are not limited to better sults for scale-dependent, bg(k), redshift-dependent, bg(z), control of galaxy bias uncertainty. and both scale- and redshift-dependent galaxy bias, bg(k,z). More grid nodes means more flexibility and reduces the con- 5.3.4 rg, the cross-correlation coefficient straining power of our spec-z survey. The bg(k,z) results stop at nk = nz = 5 for computational reasons. Our implemen- Just as galaxy bias, bg(k,z), is a nuisance parameter which tation is based on that of Joachimi & Bridle (2010). describes our ignorance of the extent to which galaxies are a The top-left panel shows results for the spec-z nn sur- biased tracer of dark matter, there is an analogous term in vey alone. FoM decreases as we increase the flexibility of the the nǫ observable which appears where we cross-correlate grid. While FoM falls quickly to very low levels for bg(k,z) galaxy clustering and cosmic shear. The term, which we & bg (z), the bg(k) FoM retains roughly half it’s max value refer to here as the galaxy-shear cross-correlation coeffi- by nk = 2 and falls slowly thereafter, reaching 1/3 of its cient, rg(k,z), is a measure of the statistical coherence of maximum value by nk = 10. It’s possible that this relative the two fields (Baldauf et al. 2010; Guzik & Seljak 2001; insensitivity to an increaingly scale-dependent redshift term Mandelbaum et al. 2013). rg = 1 means the fields (galaxy is due to the stringent ell-cuts we impose on our spec-z sur- and matter overdensities respectively) are fully correlated, vey, removing non-linear and quasi-linear scales. there is a deterministic mapping between the two fields. Interestingly, there is a relatively consistent plateau Gazta˜naga et al. (2012) have plausibly argued that, when above nk = nz = 2 for all bias types and all probe combi- we restrict ourselves to linear scales, where galaxy bias can nations, suggesting our fiducial 2 2 grid approach is a sen- be assumed to be broadly scale-independent, then we can × sible choice if we are not to over-constrain cosmology from assume rg(k,z) = 1. Cacciato et al. (2012) found rg 1 on ≈ LSS. A higher resolution grid is computationally more in- large scales based on the halo model. c 2009 RAS, MNRAS 000, 1–21 14 Donnacha Kirk, Ofer Lahav, Sarah Bridle, Stephanie Jouvel, Filipe B. Abdalla, Joshua A. Frieman nn εε + nn 8 20 b(k) b(z) 6 b(k,z) 15 4 10 FoM FoM 2 5 0 0 0 2 4 6 8 10 0 2 4 6 8 10 n = n n = n k z k z εε + nε + nn 50 7 40 6 5 30 4 FoM 20 3 same−sky benefit 10 2 0 1 0 2 4 6 8 10 0 2 4 6 8 10 n = n n = n k z k z Figure 4. DE FoM as a function of the number of grid nodes in k/z space for our galaxy bias model. Fiducial surveys and forecast assumptions assumed. Results are shown for nn-only [top left], ǫǫ+nn [top right] and ǫǫ+nǫ+nn [bottom left]. We also show the relative improvement due to a same sky analysis [bottom right] i.e. the ratio of F oMǫǫ+nǫ+nn to F oMǫǫ+nn. Each panel presents results for bias which depends only on redshift, bg(z), [red dashed], bias which depends only on scale, bg(k), [blue dotted], and bias which depends on both scale and redshift, bg (k,z), [black solid]. nk and nz are the number of nodes in k/z space. These vary around their fiducial value of one and are interpolated to give a bg “surface” in k/z. z-nodes are linearly spaced, k-nodes are log-spaced. In this section we relax this assumption, parameterising learn more about the photometric redshift distribution than rg(k,z) in the same way as we treat bg(k,z), i.e. one free am- we can using the photo-z survey alone. plitude term and a 2 2 grid in k/z-space, and marginalising × In practice the extra specroscopic redshift information the resulting five nuisance parameters. could be fundamentally integrated into the calibration of the Only the cross-correlation term, nǫ, is sensitive to photo-z sample. Here we take a more general approach pa- rg(k,z). As such, marginalisation over this extra term only rameterise the error on our photometric n(z) in some way, effects our same-sky combination of probes, ǫǫ + nǫ + nn, then allow these parameters to vary as new nuisance param- reducing the benefit from a same sky analsysis. Assuming, eters. Extra photo-z ‘’calibration” from the addition of the as we do here, that rg is as strong a contaminant as bg is spec-z survey enters as tighter constraints on these photo- a rather pessimistic scenario, strongly penalising same-sky z nuisance parameters. This information is only available coverage. Even so there is still a 25% improvement when ∼ when the two surveys overlap so it forms a contribution to- same-sky constraints are compared to the independent com- wards the same-sky benefit we observe. bination of surveys. In our fiducial model we use a single global parame- ter δz as our photo-z nuisance parameter. It enters into the overall photo-z error as σz = δz(1 + z) and we allow it to vary around our DES-like fiducial value of δz = 0.07. One 5.4 Photometric Redshift Error photo-z nuisance parameter is relatively conservative but as As well as improvements in the ability to constrain cos- there have been some reservations expressed about the effi- mology and control for galaxy bias, there has been much cacy of this kind of “self-calibration” we consider this a con- interest in the combination of photo-z and spec-z surveys servative choice. Having many photo-z nuisance parameters to “self-calibrate” the photometric redshift error (Newman weights the entire forecast methodology strongly towards a 2008; Zhang et al. 2010). The principle being invoked here poorly estimated photo-z distribution which becomes much is straightforward: if the spec-z survey offers us highly accu- improved by cross-correlation with a spec-z survey. rate redshifts for some sub-set of the galaxies in the photo-z In this section we relax some of these assumptions survey then we should be able to use this information to and investigate in more detail the impact of photo-z mis- c 2009 RAS, MNRAS 000, 1–21 Optimising Spectroscopic & Photometric Galaxy Surveys: Same-sky Benefits for Dark Energy & Modified Gravity 15 140 120 DE nn DE 120 DE εε+nn DE εε+nε+nn 100 MG 100 MG nn MG εε+nn 80 80 MG εε+nε+nn 60 FoM 60 FoM 40 40 20 20 0 −4 −2 0 2 0 10 10 10 10 0 1000 2000 3000 4000 5000 prior on δi & ∆i 2 z z overlap [deg ] 12 4 DE DE 10 MG 3 MG 8 6 2 4 same−sky benefit 1 same−sky benefit 2 0 0 −4 −2 0 2 10 10 10 10 0 1000 2000 3000 4000 5000 prior on δi & ∆i 2 z z overlap [deg ] Figure 5. This plot uses our fiducial forecast assumptions and Figure 6. [Top panel] FoMs for DE [solid] and MG [dashed] for surveys but allows photo-z error to vary more freely. We use ten our ǫǫ+nǫ+nn from our photo-z and spec-z surveys with fiducial photo-z nuisance parameters: a gaussian error per tomographic forecast assumptions as a function of survey overlap. Both surveys i 2 bin, δz , with fiducial values 0.07, and a mean redshift offset per have their fiducial 5000deg area but the nǫ cross-correlation can i 2 bin, ∆z, with fiducial values 0. [Top panel] DE [solid lines] & MG only exploit sky area for which the surveys overlap. For 0 deg [dashed lines] FoMs as a function of the prior on our photometric overlap there is no nǫ contribution. [Lower panel] Same-sky ben- i i redshift nuisance parameters, δz & ∆z. FoMs are shown for nn efit i.e. F oMǫǫ+nǫ+nn/F oMǫǫ+nn as a function of overlapping alone [black lines], ǫǫ + nn [red lines] and ǫǫ + nǫ + nn [blue lines]. area. i i [Lower panel] Same-sky benefit as a function of prior on δz & ∆z for DE [solid] and MG [dashed]. estimation on our survey constraints and the same-sky ben- efit from cross-correlation. We follow the approach of Bordoloi et al. (2010); as marginalising over the photo-z nuisance parameters has Amara & Refregier (2006) by introducing 2Nz nuisance pa- less impact. The improvement is more pronounced for MG rameters, where Nz is the number of tomographic bins used than it is for DE, suggesting that the z-dependence of the to analyse our photo-z survey (Nz = 5 for our DES-like sur- MG parameters is more degenerate with the photo-z error vey). We allow δz to vary independently in each z-bin around than that of w0,wa. The major improvement in constrain- the fiducial value of 0.07 and we introduce a new nuisance ing power comes between prior values of 0.1 and 0.01, with parameter- the bias on the mean redshift of each bin, which plateaus above and below this range. This finding concurs we allow to vary independently around zero. We allow these with that of Kirk et al. (2011). ten nuisance parameters to vary freely with wide, flat priors. As prior ranges on the photo-z nuisance parameters are This represents a case of very poor photo-z estimation. We tighterned, the same-sky benefit drops (lower panel). This is then increase the prior on all the photo-z nuisance parame- expected as one effect of the nǫ cross-correlation is to “self- ters to show the change in FoMs and same-sky benefit with calibrate” photo-z error. As the priors are tightened, the improving photo-z knowledge. Results are shown in Fig. 6. cross-correlation has less work to do so the benefit it confers In the case of wide, flat prior (right hand side of the is less. Even so, while the DE same-sky benefit reduces from plots) FoM for both DE and MG is reduced for any probe a high of a factor of 11 for a prior of width 10, it is still combination that includes the ǫǫ photo-z survey (the nn lines more than a factor of 3 with a very tight prior of width − are flat as the photo-z nuisance parameters do not impact 1e 4. This confirms that the same-sky benefit is not due this probe). Moving from right to left on the plot, the priors primarily, or even predominantly, to the self-calibration of on all the photo-z nuisance parameters are tightened. This photo-z error. In fact, for both DE and MG, the same-sky improves the constraining power of any survey including ǫǫ benefit is relatively stable below a prior of 0.01. c 2009 RAS, MNRAS 000, 1–21 16 Donnacha Kirk, Ofer Lahav, Sarah Bridle, Stephanie Jouvel, Filipe B. Abdalla, Joshua A. Frieman 5.5 Survey Overlap: Area on the Sky WGL survey the relevant quantity is the lensing efficiency function for a particular bin which tends to peak at about The same-sky benefit of overlapping a spec-z LSS and photo- half the peak redshift of the galaxy distribution. z WGL survey had been a main focus of investigation for this Fig. 7 shows the impact of changing the fiducial z-range paper. Here and in the next section we examine the results of our LSS survey, zmin = 0 c 2009 RAS, MNRAS 000, 1–21 Optimising Spectroscopic & Photometric Galaxy Surveys: Same-sky Benefits for Dark Energy & Modified Gravity 17 30 30 nn 25 25 εε + nn εε + nε + nn 20 20 15 15 DE FoM DE FoM 10 10 5 5 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 z z max min 4 4 3.5 3.5 3 3 2.5 2.5 same−sky benefit 2 same−sky benefit 2 1.5 1.5 1 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 z z max min Figure 7. This plot explores the importance of the z-coverage of our spec-z survey on individual and combined constraints. The spec-z survey n(z) is assumed to be flat between its zmin and zmin values. All other survey assumptions are fixed at their fiducial values. [Left panels] Vary survey zmax while keeping zmin = 0 fixed. [Right panels] Vary zmin while keeping zmax=1.7 fixed. [Top panels] DE FoM for spec-z nn alone [black lines], the independent combination of spec-z and photo-z, ǫǫ + nn, [red lines] and the full joint combination including cross-correlations [blue lines]. [Bottom panels] Same-sky benefit, i.e. F oMǫǫ+nǫ+nn/F oMǫǫ+nn, as a function of zmin and zmax. case because, without strong RSDs. It does not make sense 5.8 Non-linear Scales to discuss a “same-sky benefit” in this case because we are In the preceding sections we have assumed a cut which re- dealing with datasets from the same photo-z survey, never- moves modes from our LSS analysis corresponding to non- theless we can say that there is strong improvement when linear and quasi-linear scales. The prescription we use is the nǫ correlations are included, producing a DE FoM nearly taken from Rassat et al. (2008), section 4.3. This approach as strong as that from the photo-z + spec-z case. For MG removes scales smaller than l = k χ(zi ), where the final constraint is much less strong, giving a MG FoM max max med χ(zi ) is the comoving distance of the median redshift of less than a quarter the size of that achieved by photo-z + med tomographic bin i, i.e. the scale-cut is redshift dependent spec-z. We can assert that, for DE, increased number density by bin. k is defined by considering only scales for which makes up for lower z-resolution/reduced RSD effects but the max σ(R) < X where MG constraint suffers from the lack of RSDs which reduces the orthogonality of the WGL and LSS data. dk 9 σ2(R)= ∆2(k) [sin(kR) kRcos(kR)]2 . (29) Z k (kR)6 − Peacock & Dodds (1996) eqn. 42 defines a R value and k = 2π/R relates this to a kmax. Our fiducial choice is X = 0.2, −1 corresponding to kmax 0.25hMpc . Figure 8 shows the ∼ In conclusion, the combination of our fiducial photo-z impact of changing this assumption. We show the change in and spec-z surveys outperforms the joint WGL + LSS con- FoMs and same-sky benefit from varying the X value which, straints from the photo-z survey alone, particularly in con- in turn, changes the maximum k-value which we include in straining deviations from GR. This is true even though our our nn and nǫ C(l)s. fiducial spec-z survey has shallower z-coverage and a signifi- As expected, including more non-linear scales in our cantly lower number density than our photo-z survey. While analysis improves FoM from all forecasts which include LSS any complete analysis will exploit LSS information from the observables. For nn alone, when we include smaller scales, photo-z survey, where a suitable spec-z survey is available, the nn survey quickly overtakes the constraining power of joint constraints between surveys are strongly encouraged. the ǫǫ survey (shown as a horizontal line for reference, it c 2009 RAS, MNRAS 000, 1–21 18 Donnacha Kirk, Ofer Lahav, Sarah Bridle, Stephanie Jouvel, Filipe B. Abdalla, Joshua A. Frieman 140 when we fixboth bg and rg, the ǫǫ + nǫ + nn constraint in- creases strongly with large kmax but quickly plateaus above 120 kmax > 1, while nn and ǫǫ + nn continue to increase. 100 nn One constant in all our results is the rapid increase εε+nn in the constraining power of ǫǫ + nǫ + nn as we increase 80 εε+nε+nn εε our fiducial kmax, leading to a rapid improvement in same- FoM 60 sky benefit. We are content then that our choice of fiducial kmax(z = 0) 0.25 is a sensibly conservative one. Improve- ∼ 40 ments in our understanding of non-linear galaxy and DM clustering are always to be welcomed and may improve the 20 constraining power of our LSS observables but they are not 0 essential to see strong improvements in constraining power 0 0.5 1 1.5 2 2.5 3 3.5 k (z=0) [hMpc−1] max from the overlap of spec-z and photo-z surveys. 9 8 DE marg. b ,r g g 7 6 DISCUSSION DE marg. b , fix r g g 6 DE fix b ,r This paper, like others before it (Bernstein & Cai 2011; g g MG marg. b ,r Gazta˜naga et al. 2012; Cai & Bernstein 2012; Duncan et al. 5 g g MG marg. b , fix r 2013), has demonstrated the benefits of combining a photo- g g 4 MG fix b ,r metric WGL survey with a spectroscopic LSS survey. When same−sky benefit g g 3 their independent likelihoods are simply added we see more than a factor of four improvement in DE FoM on the best ei- 2 ther can do alone. This improvement is particularly marked 1 in a two parameter modified gravity model where con- 0 0.5 1 1.5 2 2.5 3 3.5 k (z=0) [hMpc−1] max straints from each probe are orthogonal and their combina- tion breaks an important degeneracy, here the ǫǫ+nn combi- nation performs more than two orders of magnitude better Figure 8. This figure shows the impact of changing the scale than either probe alone. It is worth noting that this very at which we exclude non-linear clustering from our nn and nǫ strong improvement is only visible when one goes beyond analysis. The x-axis is the maximum k value included (at z=0). the common γ parameterisation of modified linear growth We alter the k-cuts according to the recipe of Rassat et al. (2008) to a two (or more) parameter modified gravity model which where we vary the X parameter away from its fiducial value of 0.2, is sensitive to the fact that the WGL and LSS observables equivalent to k ∼ 0.25hMpc−1 on this plot. [Upper panel] FoM as are sensitive to different combinations of the metric poten- a function of the maximum k-mode included in the spec-z analysis tials. for nn [black], ǫǫ+nn [red] and ǫǫ+nǫ+nn [blue]. ǫǫ [black dash] Beyond this we have used a simple combined probes for- FoM is shown for comparison, it is never subjected to a k-cut. malism based on projected angular power spectra to calcu- [Lower panel] Same-sky benefit, i.e. F oMǫǫ+nǫ+nn/F oMǫǫ+nn, late the cross-power spectra between our photo-z and spec-z as a function of maximum k-mode included in LSS analysis. Same-sky benefit is shown for DE [solid] and MG [dashed] for surveys and also their full joint covariance matrix. The inclu- sion of these cross-correlations models the extra data avail- the cases where both bg and rg are marginalised over [blue], bg is marginalised over but rg is fixed [red] and both bg and rg are able when we have both of these cosmic probes observed on held fixed [green]. the same patch of sky. Having overlapping surveys of this nature provides a range of benefits. Photometry is neces- sary to construct a target list for spectroscopy, while spec- troscopic data can help calibrate the photometric redshift never gets cut on ell), suggesting that our NL-cut exerts distribution. Systematic effects such as galaxy bias or In- a strong constraint on the LSS observable. We see similar trinsic Alignments are often more accurately characterised improvement for the ǫǫ + nn combination, with both ex- from multiple overlapping datasets. In this paper we have hibiting plateaus above k 1 1.5hMpc−1. In contrast the concentrated on the improved constraints on cosmology and ∼ − full combination of WGL & LSS surveys, including cross- nuisance parameters that come from conducting a full joint correlations, ǫǫ + nǫ + nn shows a very strong improvement likelihood calculation in our C(l)s formalism. in FoM right down to k = 3hMpc−1. In this case increas- This has allowed us to define a same-sky benefit factor- −1 −1 ing kmax(z = 0) from 0.25hMpc to 3hMpc improves the improvement when these nǫ cross-correlations are in- FoM by a factor of four. We would not have necessarily ex- cluded. Four our fiducial forecast assumptions we see strong pected this behaviour but it seems to suggest that the inclu- positive same-sky improvements of nearly a factor of four sion of highly nonlinear scales combined with the WGL/LSS for DE and more than a factor of two for MG. cross-correlation breaks cosmological parameter degenera- Any such forecast is a complicated calculation, within cies more fundamental than the simple control of bg. It which many assumptions are made which can radically affect could simply be due to the fact that the cross-correlation the final results. We have tried to methodically disentangle combination has another set of nuisance parameters to con- a number of the most important assumptions in an effort to strain (rg) that continue to benefit from NL scales, even quantify their impact and produce the most robust range of after nn itself has exhausted its constraining power. Indeed forecasts possible. c 2009 RAS, MNRAS 000, 1–21 Optimising Spectroscopic & Photometric Galaxy Surveys: Same-sky Benefits for Dark Energy & Modified Gravity 19 We are confident that the general trend of our fiducial WGL lensing kernel. Once this requirement has been met, survey results are robust to the inclusion of priors from a pushing to higher redshift and thus increased volume for the Planck-like CMB experiment and for the type of feature-full spec-z survey continues to be very beneficial. spec-z n(z) produced by any specific spectrograph/telescope On the issue of same-sky improvement, the benefit from combination, target selection choices and survey strategy. the extra cross-correlations available when our LSS and These choices are investigated in more detail in our com- WGL surveys overlap on the sky, we can see a range of re- panion paper Jouvel (2013). sults depending on a variety of assumptions that are made The importance of galaxy-shear cross correlations when making forecasts. Clearly the worst same-sky benefit for controlling galaxy bias has been extensively noted results come from aggressively marginalising an unknown (Yoo & Seljak 2012; de Putter et al. 2013; Asorey et al. galaxy-shear cross-correlation, rg, which results in a same- 2013). We show that indeed choices of galaxy bias model, sky benefit factor of 1.2. This is highly pessimistic and there nuisance parameterisation and galaxy population bias am- are strong arguments, both theoretical (Gazta˜naga et al. plitude can all significantly effect both LSS-only constraints 2012) and observational (Comparat 2013b), that rg is very and combined WGL+LSS constraints. Galaxy bias mod- close to unity on the linear scales for which we consider LSS elling is an area of active research interest which will benefit data. greatly from improved observations over the coming years. For some assumptions we see very strong same-sky im- Currently the state of our knowledge of bg is limited enough provements. Most promisingly the example we take of a n(z) the necessitate the inclusion of nuisance parameters which, based on a specific target selection and survey strategy sce- when marginalised over, are a way of including our uncer- nario shows a DE same-sky benefit of more than a factor of tainty about the true galaxy bias into a forecast. More nui- ten. While this number will be very dependent on the de- sance parameters decreases our ability to measure cosmology tails of target selection, survey strategy etc, it is still very but will produce a more robust, less biased final result. We promising when we consider the application of this analy- remove truly non-linear scales (for which bg modelling is par- sis to real survey data. In addition, the targetting of more ticularly uncertain) from our analysis entirely with judicious strongly biased galaxy populations, not only makes the LSS cuts on small scales. probes more constraining but increases the same-sky factor. We show that, while increasing the number of galaxy This is clearly of relevance to the McDonald & Seljak (2009) bias nuisance parameters does reduce our constraining technique for the control of cosmic variance and we intend power, there is little decrease beyond a 4 4 grid of nui- to produce a more comprehensive analysis in a future paper. × sance parameters in k/z space, i.e. a bias model with 17 free In general we see that DE and MG follow very similar parameters. The same-sky benefit does increase with more trends as we perturb our assumptions away from the fiducial uncertainty in galaxy bias, supporting the assertion that the model. The major difference remains the fact that MG ben- nǫ correlation can control for bg, however, even if we assume efits so strongly from the independent combination of WGL bias is known perfectly there is a factor of three benefit from and LSS due to the orthogonality of the constraint contours. the extra correlations offered by overlapping surveys. This produces such a dramatic improvement that the same- A similar effect is observed when we increase the uncer- sky benefit is correspondingly less pronounced than in the tainty in the photometric redshift error. As the nǫ correla- DE case. We see lower same-sky benefits for MG than for tion can go some way towards “calibrating” this error, there DE for all forecast assumptions (with the exception of the is more scope for improvement when the photo-z error is less case without RSDs which is only included to demonstrate well understood and the same-sky benefit is correspondingly the power of the use of RSDs). Nevertheless the MG same- higher. However, we want to emphasis that we find, as with sky benefit remains at the level of a factor of two or more galaxy bias, that there is still substantial improvement due for most sensible forecast assumptions. to nǫ cross-correlations even in the case where the photo-z There are a range of possible extensions to the work we error is assumed to be perfectly described. present in this paper. The joint analysis of different cosmo- The forecasting assumption that most impacts the logical data sets is becoming more ambitious. We hope to same-sky benefit is our assumed knowledge of the galaxy- extend our C(l)s formalism to allow the cross-correlation of shear cross-correlation coefficient, rg. If we allow this to vary an arbitrarily large number of observables in any bin combi- with the same freedom as our fiducial bg model then same- nations. Among other things this would allow us to include sky benefit is reduced to less than a factor of 1.2. However LSS information from our photo-z survey as well as break there are strong theoretical arguments that suggest rg is our galaxy populations up into population samples which are close to unity, at least on the linear and quasi-linear scales differently biased. McDonald & Seljak (2009) suggest that we include here. We suggest that the very low same-sky ben- this is an effective way to reduce cosmic variance. We also efits found from aggressive marginalisation over rg are overly aim to include cosmic magnification in our future efforts as pessimistic (Gazta˜naga et al. 2012). well as conduct a more detailed study of the trade-off in As well as assessing the difference between combined accuracy due to projection effects when modelling a spec-z constraints from surveys on different parts of the sky versus galaxy survey in projected angular power spectra. surveys which completely overlap, we also look at the effects of partial overlap, both in area on the sky and in z-coverage. We clearly see that even a partial overlap is beneficial, par- 7 CONCLUSIONS ticularly for MG, where half the full 5000 deg2 overlap ben- efit comes from the first 1000 deg2. Our z-coverage analy- Joint survey analysis is clearly an essential part of cosmology sis shows two complementary sources of improvement. Most if we are to make the most of the unprecedented data sets important is that the spec-z survey covers the peak of the shortly to become available from a range of cosmic probes. c 2009 RAS, MNRAS 000, 1–21 20 Donnacha Kirk, Ofer Lahav, Sarah Bridle, Stephanie Jouvel, Filipe B. Abdalla, Joshua A. Frieman Different probes go beyond the sum of their parts through where have only considered one observable from each of our the breaking of degeneracies between cosmological parame- two surveys, it is clear that the range of assumptions that ters. In addition they can help to constrain systematic ef- go into the forecast make the prediction a complex one. As fects and unknown physical quantities e.g. halo masses or the number of observables included in a simultaneous joint cluster-mass relations. Multiple probes on the same patch analysis are increased this effect will only become more pro- of sky often observe the same objects and there is obvious nounced. It is vitally important that all assumptions are synergy between photometric and spectroscopic surveys in stated explicitly and examined in isolation to determine terms of target selection and photo-z error calibration. their effect relative to others. If we conduct this process We have produced a range of joint forecasts, combining correctly the prize is enormous: highly precise cosmological generic photometric WGL and spectroscopic LSS surveys. measurements, far beyond anything available to probes con- Even a simple forecast of this kind requires a large number sidered in isolation. Only in this way will we be fully able of assumptions, often implicit. We have tried to lay bare to exploit our available data and more precision cosmology every part of the process and conduct a sensitivity analysis onto the next level. by varying each assumption in turn and quantifying their impact on individual and combined constraints relative to eachother. ACKNOWLEDGEMENTS Throughout our sensitivity analysis we see some con- stant trends: (i) the combination of WGL from a photo-z OL acknowledges a Royal Society Wolfson Research Merit survey and LSS from a spec-z survey greatly improve our Award, a Leverhulme Senior Research Fellowship and an ability to measure the equation of state of DE and deviations Advanced Grant from the European Research Council. FBA from General Relativity, by a factor of four in DE FoM for thanks the Royal Society for support via an URF. SB ac- our fiducial surveys compared to photo-z WGL alone, (ii) in knowledges support from European Research Council in the the MG case in particular the orthogonal nature of the con- form of a Starting Grant with number 240672. straints from both probes produces a very strong joint con- Many thanks to all those who engaged in very helpful straint once degeneracies in the MG parameters are broken, discussions on the combined probes and same sky issues, es- improving our MG FoM by over two orders of magnitude for pecially Enrique Gaztanaga, Jacobo Asorey, Gary Bernstein non-overlapping surveys compared to photo-z WGL alone, and Yan-Chuan Cai. We are indebted to the DESpec col- and (iii) there is a significant benefit from overlapping sur- laboration for useful discussion and collaboration. Thanks veys on the same patch of sky which allows us access to the to Jochen Weller for supplying the Planck FM and Tom nǫ cross correlation between probes and the full covariance Kitching on its relation to Planck data. Thank you to Anais matrix including all off-diagonal elements, giving an extra Rassat, Ole Host, Lisa Voigt and Lucy Clerkin for insights factor of four for DE and more than two for MG compared into nuisance parameters and forecast assumptions. to non-overlapping surveys with our fiducial assumptions. Different groups have produced conflicting results on This paper has been typeset from a TEX/ LATEX file prepared the question of the same-sky benefit from overlapping sur- by the author. veys such as those we consider. While Gazta˜naga et al. (2012) see significant improvement from same-sky overlap, Cai & Bernstein (2012) predict little improvement over the REFERENCES independent combination of WGL/LSS as if they were on different patches of sky. 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